An approximation to zeros of the Riemann zeta function using fractional calculus

TL;DR

This paper introduces a novel fractional calculus-based iterative method to approximate the zeros of the Riemann zeta function, especially for large imaginary parts, using derivatives of constant functions.

Contribution

The paper presents the first use of fractional derivatives of constants in an iterative method to find multiple zeros of the Riemann zeta function with a single initial condition.

Findings

Successfully approximated zeros near the critical line

Demonstrated the method's applicability for large imaginary parts

Provided 53 numerical examples close to the zeros

Abstract

In this document, as far as the authors know, an approximation to the zeros of the Riemann zeta function has been obtained for the first time using only derivatives of constant functions, which was possible only because a fractional iterative method was used. This iterative method, valid for one and several variables, uses the properties of fractional calculus, in particular the fact that the fractional derivatives of constants are not always zero, to find multiple zeros of a function using a single initial condition. This partly solves the intrinsic problem of iterative methods that if we want to find N zeros it is necessary to give N initial conditions. Consequently, the method is suitable for approximating nontrivial zeros of the Riemann zeta function when the absolute value of its imaginary part tends to infinity. The deduction of the iterative method is presented, some examples of its implementation, and finally 53 different values near to the zeros of the Riemann zeta function are shown.